\begin{subanswer}{iszerocorrelationindependentexamples}
Let $Z$ have the probability mass function
\begin{align*}
p(z) =
\begin{cases}
  0.5 & \text{ if } z = -1 \\
  0.5 & \text{ if } z = 1
\end{cases}
\end{align*}
or equivalently
\begin{align*}
P(Z = -1) &= 0.5  \\
P(Z =  1) &= 0.5
\text{.}
\end{align*}
Let $X \sim \text{Normal}(0,1)$
and
$Y = ZX$.
Derive the distribution of $Y$ using its cumulative distribution function:
\begin{align*}
P(Y<x) &=
P( ZX<x| Z = 1)P(Z=1) +
P( ZX<x| Z = -1)P(Z=-1) \\
&=
P( X<x)(0.5) + P(-X<x)(0.5) \\
&=  (0.5)( P( X<x) + P(X \geq -x)) \\
&=  (0.5)( P( X<x) + P(X < x)) \quad \text{symmetry} \\
&=   P(X < x)
\end{align*}
meaning $Y$ has the same distribution as $X$.
Now, check whether $\rho=0$,
\begin{align*}
\rho
&= \frac{\Cov(X,Y)}{\sigma_X \sigma_Y}  \\
&= \Cov(X,Y) \\
&= \Cov(X, ZX) \\
&= \E(XZX) - \E(X) \E(ZX)  \\
&= \E(X^2 Z) - \E(X) \E(ZX)  \\
&= \E(X^2) \E(Z) - \E(X)\E(Z)\E(X)   \quad \text{independence} \\
&=  0                                \quad \text{since } E(Z)=0
\text{.}
\end{align*}
You have successfully constructed a situation where
there is a clearly defined dependence between $Y$ and $X$, but
they are both distributed $\text{Normal}(0,1)$ with correlation coefficient
$\rho = 0$.
\end{subanswer}
